### Parsing in Erlang

This is a simple example of building a traditional parser in Erlang. The technology is recursive descent. It is essentially my first Erlang program, so I cannot make claims for Erlang quality. But it works.

The input is a simple expression language. I have built a much more elaborate version but the central idea gets lost in the clutter. The grammar is left recursive. The resulting AST must therefore show left recursive structure. The main "trick" is making this happen.

The output is an Abstract Syntax Tree expressed as a tuple. The meaning of the tree should be obvious.

The Erlang code.

```% FILE:    exprParser.erl
% PURPOSE: parse an expression
% AUTHOR:  mckeeman@mathworks.com
%
% EXAMPLES:
%   exprParser:parse("")                 ""
%   exprParser:parse("x")                "x"
%   exprParser:parse("(x)")              "x"
%   exprParser:parse("y*(x-1)")          {mpy,"y",{sub,"x",1}}
%
% CFG:
%   expr   = sum
%   sum    = term
%   sum    = sum '+' term
%   sum    = sum '-' term
%   term   = term '*' factor
%   term   = term '/' factor
%   factor = 0-9
%   factor = a-z
%   factor = '(' sum ')'
%
% The left recursion in the CFG is turned into an iteration
% in the regular expression grammar.
% REG:
%   expr   = sum
%   sum    = term ('+' term | '-' term)*
%   term   = factor ('*' factor | '/' factor)*
%   factor = 0-9 | a-z | '(' sum ')'
%
% METHOD:
%   The parser takes each character in turn.
%   The result is an Abstract Syntax Tree.
%   Constant nodes are folded.
% NODE:
%   {op,node, node} or id or int.
% PARSE TREE:
%   nested nodes
%
% IMPLEMENTATION:
%   The typical parse function f has the form
%      {Ast, text} = f(Ast, text)
%   where at each step the Ast grows, the text becomes shorter.
%   Syntax errors yield cryptic failure messages.

-module(exprParser).
-export([parse/1]).

parse("")  -> "";
parse(Txt) ->
{Ast, []} = expr(Txt),
Ast.

expr(Txt) -> sum(Txt).

% sum = term ('+' term | '-' term)*
sum(Txt) ->
{Ast, Txt1} = term(Txt),                       % at least one term
addops(Ast, Txt1).                             % the rest, if any

{Ast1, Txt1} = term(Txt),
{Ast1, Txt1} = term(Txt),
Ast2 = operate(sub, Ast, Ast1),
addops(Ast, Txt) -> {Ast, Txt}.                  % no more to do

% term = factor ('*' factor | '/' factor)*
term(Txt) ->
{Ast, Txt1} = factor(Txt),                     % at least one factor
mulops(Ast, Txt1).                             % the rest, if any

mulops(Ast, []) -> {Ast, []};
mulops(Ast, [\$*|Txt]) ->                         % *
{Ast1, Txt1} = factor(Txt),
Ast2 = operate(mpy, Ast, Ast1),
mulops(Ast2, Txt1);
mulops(Ast, [\$/|Txt]) ->                         % /
{Ast1, Txt1} = factor(Txt),
Ast2 = operate(dvd, Ast, Ast1),
mulops(Ast2, Txt1);
mulops(Ast, Txt) -> {Ast, Txt}.                  % no more to do

% factor = '(' sum ')' | a-z | 0-9
factor([\$(|Txt]) ->                              % ( sum )
{Ast, [\$) | Txt1]} = sum(Txt),                 % discard )
{Ast, Txt1};
factor([H|T]) when H>=\$a, \$z>=H -> {[H], T};     % lower case letters
factor([H|T]) when H>=\$0, \$z>=9 -> {H-\$0, T}.    % digits

% build subexpression node or constant fold
operate(Op, V1, V2) ->
if
is_number(V1) andalso is_number(V2) ->       % both args const
case Op of                                 % Erlang arith